Fundamentals of probability

02/18/2020

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Outline

  • The following topics will be covered in this lecture:
    • Events
    • Sample spaces
    • Classical version of probability
    • Relative frequency approximation of probability
    • Probabilistic reasoning
    • Complements of events
    • Odds

Basics of probability

  • In order to interpret a population from a sample, we required that the sample was representative of the full population.

    • Using random chance to mix the participants in the samples was one way of ensuring that the smaller sample would be a good approximation of the population.

  • However, in any sample there is natural variation amongst those sampled that leads to two repeated samples not to look like eachother.

    • One of the important considerations is thus, how likely would it be to compute a sample statistic just by chance, due to the natural variation of resampling?

  • Inferential statistics are differentiated from the descriptive statistics we have seen so far in how they address this above question.

    • Descriptive statistics help us learn about the sample we have in hand, but we must use tools from probability to address how these results might generalize to a wider population.

  • One of the most important tools we will learn in this class is hypothesis testing, as one way to test if a claim might be infered about the wider population.

Basics of probability example

Diagram of hypothesis testing for a gender selection method.

Courtesy of Mario Triola, Essentials of Statistics, 6th edition

  • We can take a basic example to illustrate this point.
  • Suppose a clinical trial will be used to determine if a certain fertility treatment will increase the chance that a pregnancy will result in a female birth.
  • There is some random chance involved in an un-assisted pregnancy whether the baby will be a girl or boy, and either result is about as likely as the other.
    • Therefore, every group of \( 100 \) births in a control group might look quite different.
  • We want to find a way to be more confident that if there is an effect of the treatment, it can be distinguished from this random variation.
  • Formally we will make a claim and a hypothesis:
    • Claim: the fertility treatment will greatly increase the chance of a baby being born a girl over the control group.
    • Hypothesis: the treatment has no effect and it is equally likely that a pregnancy will result in a female or a male birth under the treatment.
  • Note: in the above, we take the null hypothesis, i.e., to test the claim we assume that it is not true and then evaluate the results.

Basics of probability example